A220264 -id:A220264 - cp's OEIS frontend

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310

Offset: 1

David W. Wilson

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.
A111059 is a subsequence. - Reinhard Zumkeller, Jul 05 2010
Choie et al. (2007) call these "Hardy-Ramanujan integers". - Jean-François Alcover, Aug 14 2014
The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".
For all such sequences b for which it holds that b(n) = b(A046523(n)), the sequence which gives the indices of records in b is a subsequence of this sequence. For example, A002182 which gives the indices of records for A000005, A002110 which gives them for A001221 and A000079 which gives them for A001222. - Antti Karttunen, Jan 18 2019
The prime signature corresponding to a(n) is given in row n of A124832. - M. F. Hasler, Jul 17 2019

			The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (first 291 terms from Will Nicholes)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017. See section 8.2, "Hardy-Ramanujan Numbers".
YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol. 19, No. 2 (2007), pp. 357-372. See section 5, p. 367.
Asaf Cohen Antonir and Asaf Shapira, An Elementary Proof of a Theorem of Hardy and Ramanujan (2022). arXiv:2207.09410 [math.NT]
Michael De Vlieger, Relations of A025487 to A002110, A002182, and A002201.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, pp. 9-10.
G. H. Hardy and S. Ramanujan, Asymptotic formulae for the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132. Also published in the book Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245-261.
Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers, Journal of Number Theory, Vol. 8, No. 4 (1976), pp. 372-389. See page 388.

Subsequence of A055932, A191743, and A324583.
Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.
Cf. A085089, A101296 (left inverses).
Equals range of values taken by A046523.
Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).
Subsequences of this sequence include: A000079, A000142, A000400, A001013, A001813, A002110, A002182, A005179, A006939, A025527, A056836, A061742, A064350, A066120, A087980, A097212, A097213, A111059, A119840, A119845, A126098, A129912, A140999, A166338, A166470, A166472, A166473, A166475, A167448, A168262, A168263, A168264, A179215, A181555, A181804, A181806, A181809, A181818, A181822, A181823, A181824, A181825, A181826, A181827, A182763, A182862, A182863, A212170, A220264, A220423, A250269, A250270, A260633, A266047, A284456, A300357, A304938, A329894, A330687; also A037019 and A330681 (when sorted), possibly also A289132.
Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.
Cf. also array A124832 (row n = prime signature of a(n)) and A304886, A307056.

import Data.Set (singleton, fromList, deleteFindMin, union)
a025487 n = a025487_list !! (n-1)
a025487_list = 1 : h [b] (singleton b) bs where
   (_ : b : bs) = a002110_list
   h cs s xs'@(x:xs)
     | m <= x    = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'
     | otherwise = x : h (x:cs) (s  `union` fromList (map (* x) (x:cs))) xs
     where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 06 2013

isA025487 := proc(n)
    local pset,omega ;
    pset := sort(convert(numtheory[factorset](n),list)) ;
    omega := nops(pset) ;
    if op(-1,pset) <> ithprime(omega) then
        return false;
    end if;
    for i from 1 to omega-1 do
        if padic[ordp](n,ithprime(i)) < padic[ordp](n,ithprime(i+1)) then
            return false;
        end if;
    end do:
    true ;
end proc:
A025487 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        1 ;
    else
        for a from procname(n-1)+1 do
            if isA025487(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A025487(n),n=1..100) ; # R. J. Mathar, May 25 2017

PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, 2350}]; ln (* Robert G. Wilson v, Aug 14 2004 *)
(* Second program: generate all terms m <= A002110(n): *)
f[n_] := {{1}}~Join~
  Block[{lim = Product[Prime@ i, {i, n}],
   ww = NestList[Append[#, 1] &, {1}, n - 1], dec},
   dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];
   Map[Block[{w = #, k = 1},
      Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],
        Product[Prime@ i, {i, Length@ w}] ] &@ Reap[
         Do[
          If[# < lim,
             Sow[#]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,
             If[k == 1,
               MapAt[# + 1 &, w, k],
               PadLeft[#, Length@ w, First@ #] &@
                 Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],
           {i, Infinity}] ][[-1]]
] &, ww]]; Sort[Join @@ f@ 13] (* Michael De Vlieger, May 19 2018 *)

isA025487(n)=my(k=valuation(n,2),t);n>>=k;forprime(p=3,default(primelimit),t=valuation(n,p);if(t>k,return(0),k=t);if(k,n/=p^k,return(n==1))) \\ Charles R Greathouse IV, Jun 10 2011

factfollow(n)={local(fm, np, n2);
  fm=factor(n); np=matsize(fm)[1];
  if(np==0,return([2]));
  n2=n*nextprime(fm[np,1]+1);
  if(np==1||fm[np,2]Franklin T. Adams-Watters, Dec 01 2011 */

is(n) = {if(n==1, return(1)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2],,4) == f[, 2]} \\ David A. Corneth, Feb 14 2019

upto(Nmax)=vecsort(concat(vector(logint(Nmax,2),n,select(t->t<=Nmax,if(n>1,[factorback(primes(#p),Vecrev(p)) || p<-partitions(n)],[1,2]))))) \\ M. F. Hasler, Jul 17 2019

\\ For fast generation of large number of terms, use this program:
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A025487list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); vecsort(Vec(lista)); }; \\ Returns a list of terms up to the term 2^e.
v025487 = A025487list(101);
A025487(n) = v025487[n];
for(n=1,#v025487,print1(A025487(n), ", ")); \\ Antti Karttunen, Dec 24 2019

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
N = 2310
nmax = 2^floor(log(N,2))
sorted([j for j in (prod(sharp_primorial(t[0])^t[1] for k, t in enumerate(factor(n))) for n in (1..nmax)) if j <= N])
# Giuseppe Coppoletta, Jan 26 2015

What can be said about the asymptotic behavior of this sequence? - Franklin T. Adams-Watters, Jan 06 2010
Hardy & Ramanujan prove that there are exp((2 Pi + o(1))/sqrt(3) * sqrt(log x/log log x)) members of this sequence up to x. - Charles R Greathouse IV, Dec 05 2012
From Antti Karttunen, Jan 18 & Dec 24 2019: (Start)
A085089(a(n)) = n.
A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]
A001221(a(n)) = A061395(a(n)) = A061394(n).
A007814(a(n)) = A051903(a(n)) = A051282(n).
a(A101296(n)) = A046523(n).
a(A306802(n)) = A002182(n).
a(n) = A108951(A181815(n)) = A329900(A181817(n)).
If A181815(n) is odd, a(n) = A283980(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).
(End)
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A002110(n)) = A161360. - Amiram Eldar, Oct 20 2020

Offset corrected by Matthew Vandermast, Oct 19 2008

Minor correction by Charles R Greathouse IV, Sep 03 2010

A086971 Number of semiprime divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 3, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 2, 1, 1, 3, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 1, 3, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 4, 1, 2, 1, 1, 1, 2, 0, 2, 2, 3, 0, 3

Offset: 1

Reinhard Zumkeller, Sep 22 2003

nonn

Inverse Moebius transform of A064911. - Jonathan Vos Post, Dec 08 2004

G. H. Hardy and E. M. Wright, Section 17.10 in An Introduction to the Theory of Numbers, 5th ed., Oxford, England: Clarendon Press, 1979.

T. D. Noe, Table of n, a(n) for n = 1..10000
E. A. Bender and J. R. Goldman, On the Applications of Mobius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, (1975), 789-803.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Moebius Transform.

Cf. A001358, A064911, A001221, A000005, A000010, A004018, A007913, A056170, A079275, A001222, A220264 (least inverse).

a086971 = sum . map a064911 . a027750_row
-- Reinhard Zumkeller, Dec 14 2012

a:= proc(n) local l, m; l:=ifactors(n)[2]; m:=nops(l);
       m*(m-1)/2 +add(`if`(i[2]>1, 1, 0), i=l)
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jul 18 2013

semiPrimeQ[n_] := PrimeOmega@ n == 2; f[n_] := Length@ Select[Divisors@ n, semiPrimeQ@# &]; Array[f, 105] (* Zak Seidov, Mar 31 2011 and modified by Robert G. Wilson v, Dec 08 2012 *)
a[n_] := Count[e = FactorInteger[n][[;; , 2]], ?(# > 1 &)] + (o = Length[e])*(o - 1)/2; Array[a, 100] (* _Amiram Eldar, Jun 30 2022 *)

/* The following definitions of a(n) are equivalent. */
a(n) = sumdiv(n,d,bigomega(d)==2)
a(n) = f=factor(n); j=matsize(f)[1]; sum(m=1,j,f[m,2]>=2) + binomial(j,2)
a(n) = f=factor(n); j=omega(n); sum(m=1,j,f[m,2]>=2) + binomial(j,2)
a(n) = omega(n/core(n)) + binomial(omega(n),2)
/* Rick L. Shepherd, Mar 06 2006 */

a(n) = A106404(n) + A106405(n). - Reinhard Zumkeller, May 02 2005
a(n) = omega(n/core(n)) + binomial(omega(n),2) = A001221(n/A007913(n)) + binomial(A001221(n),2) = A056170(n) + A079275(n). - Rick L. Shepherd, Mar 06 2006
From Reinhard Zumkeller, Dec 14 2012: (Start)
a(n) = Sum_{k=1..A000005(n)} A064911(A027750(n,k)).
a(A220264(n)) = n and a(m) <> n for m < A220264(n); a(A008578(n)) = 0; a(A002808(n)) > 0; for n > 1: a(A102466(n)) <= 1 and a(A102467(n)) > 1; A066247(n) = A057427(a(n)). (End)
G.f.: Sum_{k = p*q, p prime, q prime} x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 25 2017

Entry revised by N. J. A. Sloane, Mar 28 2006

A065119 Numbers k such that the k-th cyclotomic polynomial is a trinomial.

Original entry on oeis.org

3, 6, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, 192, 216, 243, 288, 324, 384, 432, 486, 576, 648, 729, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2187, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6561, 6912, 7776, 8748, 9216

Offset: 1

Len Smiley, Nov 12 2001

nonn

Appears to be numbers of form 2^a * 3^b, a >= 0, b > 0. - Lekraj Beedassy, Sep 10 2004
This is true: see link "Cyclotomic trinomials". - Robert Israel, Jul 14 2015
3-smooth numbers (A003586) which are not powers of 2 (A000079). - Amiram Eldar, Nov 10 2020
These are the conjugates of semiprimes, where conjugation is A122111; or Heinz numbers of conjugates of length-2 partitions. - Gus Wiseman, Nov 09 2023
A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 13 2024

			The 54th cyclotomic polynomial is x^18 - x^9 + 1 which is trinomial, so 54 is in the sequence.
From _Gus Wiseman_, Nov 09 2023: (Start)
The terms and conjugate semiprimes, showing their respective Heinz partitions, begin:
    3: (2)              4: (1,1)
    6: (2,1)            6: (2,1)
    9: (2,2)            9: (2,2)
   12: (2,1,1)         10: (3,1)
   18: (2,2,1)         15: (3,2)
   24: (2,1,1,1)       14: (4,1)
   27: (2,2,2)         25: (3,3)
   36: (2,2,1,1)       21: (4,2)
   48: (2,1,1,1,1)     22: (5,1)
   54: (2,2,2,1)       35: (4,3)
   72: (2,2,1,1,1)     33: (5,2)
   81: (2,2,2,2)       49: (4,4)
   96: (2,1,1,1,1,1)   26: (6,1)
(End)

Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 733, pp. 74 and 310, Ellipses Paris, 2004.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Cyclotomic trinomials

Differs at the 18th term from A063996.
Cf. A003586, A033845, A206787.
For primes (A008578) we have conjugates A000079.
For triprimes (A014612) we have conjugates A080193.
A001358 lists semiprimes, squarefree A006881, complement A100959.
Cf. A000040, A000720, A001248, A046682, A056239, A086971, A122111, A220264.

with(numtheory): a := []; for m from 1 to 3000 do if nops([coeffs(cyclotomic(m,x))])=3 then a := [op(a),m] fi od; print(a);

max = 5000; Sort[Flatten[Table[2^a 3^b, {a, 0, Floor[Log[2, max]]}, {b, Floor[Log[3, max/2^a]]}]]] (* Alonso del Arte, May 19 2016 *)

isok(n)=my(vp = Vec(polcyclo(n))); sum(k=1, #vp, vp[k] != 0) == 3; \\ Michel Marcus, Jul 11 2015

list(lim)=my(v=List(),N); for(n=1,logint(lim\1,3), N=3^n; while(N<=lim, listput(v,N); N<<=1)); Set(v) \\ Charles R Greathouse IV, Aug 07 2015

A206787(a(n)) = 4. - Reinhard Zumkeller, Feb 12 2012
a(n) = A033845(n)/2 = 3 * A003586(n). - Robert Israel, Jul 14 2015
Sum_{n>=1} 1/a(n) = 1. - Amiram Eldar, Nov 10 2020

Offset set to 1 and more terms from Michel Marcus, Jul 11 2015

A366740 Positive integers whose semiprime divisors do not all have different Heinz weights (sum of prime indices, A056239).

Original entry on oeis.org

90, 180, 210, 270, 360, 420, 450, 462, 525, 540, 550, 630, 720, 810, 840, 858, 900, 910, 924, 990, 1050, 1080, 1100, 1155, 1170, 1260, 1326, 1350, 1386, 1440, 1470, 1530, 1575, 1620, 1650, 1666, 1680, 1710, 1716, 1800, 1820, 1848, 1870, 1890, 1911, 1938, 1980

Offset: 1

Gus Wiseman, Nov 05 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
From Robert Israel, Nov 06 2023: (Start)
Positive integers divisible by the product of four primes, prime(i)*prime(j)*prime(k)*prime(l), i < j <= k < l, with i + l = j + k.
All positive multiples of terms are terms. (End)

			The semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5), which are not all different, so 90 is in the sequence.
The terms together with their prime indices begin:
    90: {1,2,2,3}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   270: {1,2,2,2,3}
   360: {1,1,1,2,2,3}
   420: {1,1,2,3,4}
   450: {1,2,2,3,3}
   462: {1,2,4,5}
   525: {2,3,3,4}
   540: {1,1,2,2,2,3}
   550: {1,3,3,5}
   630: {1,2,2,3,4}
   720: {1,1,1,1,2,2,3}

The complement is too dense.
For all divisors instead of just semiprimes we have A299729, strict A316402.
Distinct semi-sums of prime indices are counted by A366739.
Partitions of this type are counted by A366753, non-binary A366754.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299701 counts distinct subset-sums of prime indices, positive A304793.
A299702 ranks knapsack partitions, counted by A108917, strict A275972.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A008967, A365541, A365920, A366737, A366738, A366741, A367093, A367095, A367097.

N:= 10^4: # for terms <= N
P:= select(isprime, [$1..N]): nP:= nops(P):
R:= {}:
for i from 1 while P[i]*P[i+1]^2*P[i+2] < N do
  for j from i+1 while P[i]*P[j]^2 * P[j+1] < N do
    for k from j do
      l:= j+k-i;
      if l <= k or l > nP then break fi;
      v:= P[i]*P[j]*P[k]*P[l];
      if v <= N then
        R:= R union {seq(t,t=v..N,v)};
      fi
od od od:
sort(convert(R,list)); # Robert Israel, Nov 06 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!UnsameQ@@Total/@Union[Subsets[prix[#],{2}]]&]

These are numbers k such that A086971(k) > A366739(k).

A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).

Original entry on oeis.org

Offset: 1

Gus Wiseman, Nov 04 2023

nonn

First differs from A086971 at a(90) = 3, A086971(90) = 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The prime indices of 90 are {1,2,2,3}, with semi-sums
  3 = 1+2
  4 = 1+3 (or 2+2)
  5 = 2+3
so a(90) = 3.
Alternatively, the semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5) so a(90) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

The non-binary version is A299701.
Summing over partitions gives A366738, strict A366741.
For all sums of pairs of elements we have A367095.
Positions of first appearances are A367097.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299702 ranks knapsack partitions, counted by A108917.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000005, A000720, A001248, A008967, A117855, A304792, A304793, A365541, A365920, A366740, A366753, A367093.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Total/@Subsets[prix[n],{2}]]],{n,100}]

A366739(n) = #Set(apply(d->((f)->sum(i=1,#f~,f[i,2]*primepi(f[i,1])))(factor(d)), select(d->2==bigomega(d), divisors(n)))); \\ Antti Karttunen, Jan 20 2025

a(n) <= A086971(n). - Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A367093 Least positive integer with n more semiprime divisors than semi-sums of prime indices.

Original entry on oeis.org

1, 90, 630, 2310, 6930, 34650, 30030, 90090, 450450, 570570, 510510, 1531530, 7657650, 14804790, 11741730, 9699690, 29099070, 145495350

Offset: 0

Gus Wiseman, Nov 05 2023

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
Are all primorials after 210 included?

			The terms together with their prime indices begin:
       1: {}
      90: {1,2,2,3}
     630: {1,2,2,3,4}
    2310: {1,2,3,4,5}
    6930: {1,2,2,3,4,5}
   34650: {1,2,2,3,3,4,5}
   30030: {1,2,3,4,5,6}
   90090: {1,2,2,3,4,5,6}
  450450: {1,2,2,3,3,4,5,6}
  570570: {1,2,3,4,5,6,8}
  510510: {1,2,3,4,5,6,7}

The first part (semiprime divisors) is A086971, firsts A220264.
The second part (semi-sums of prime indices) is A366739, firsts A367097.
All sums of pairs of prime indices are counted by A367095.
The non-binary version is A367105.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299701 counts subset-sums of prime indices, positive A304793.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A008967, A117855, A304792, A365541, A365920, A366738, A366740, A366753.

nn=10000;
w=Table[Length[Union[Subsets[prix[n],{2}]]]-Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

from itertools import count
from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
def A367093(n):
    for k in count(1):
        c, a = 0, set()
        for s in (sum(p) for p in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)):
            if s not in a:
                a.add(s)
            else:
                c += 1
            if c > n:
                break
        if c == n:
            return k # Chai Wah Wu, Nov 13 2023

a(n) is the least positive integer such that A086971(a(n)) - A366739(a(n)) = n.

a(12)-a(16) from Chai Wah Wu, Nov 13 2023

a(17) from Chai Wah Wu, Nov 18 2023

A367096 Irregular triangle read by rows where row n lists the semiprime divisors of n. Alternatively, row n lists the semiprime divisors of A002808(n).

Original entry on oeis.org

4, 6, 4, 9, 10, 4, 6, 14, 15, 4, 6, 9, 4, 10, 21, 22, 4, 6, 25, 26, 9, 4, 14, 6, 10, 15, 4, 33, 34, 35, 4, 6, 9, 38, 39, 4, 10, 6, 14, 21, 4, 22, 9, 15, 46, 4, 6, 49, 10, 25, 51, 4, 26, 6, 9, 55, 4, 14, 57, 58, 4, 6, 10, 15, 62, 9, 21, 4, 65, 6, 22, 33, 4, 34

Offset: 1

Gus Wiseman, Nov 08 2023

On the first interpretation, the first three rows are empty. On the second, the first row is (4).

			The semiprime divisors of 30 are {6,10,15}, so row 30 is (6,10,15). Without empty rows, this is row 19.
Triangle begins (empty rows indicated by dots):
   1: .
   2: .
   3: .
   4: 4
   5: .
   6: 6
   7: .
   8: 4
   9: 9
  10: 10
  11: .
  12: 4,6
Without empty rows:
   1: 4
   2: 6
   3: 4
   4: 9
   5: 10
   6: 4,6
   7: 14
   8: 15
   9: 4
  10: 6,9
  11: 4,10
  12: 21

For all divisors we have A027750.
Square terms are counted by A056170.
Row sums are A076290.
Squarefree terms are counted by A079275.
Row lengths are A086971, firsts A220264.
A000005 counts divisors.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, complement A100959.
Cf. A008967, A365829, A366738, A366739, A366740, A367093, A367098.

Table[Select[Divisors[n],PrimeOmega[#]==2&],{n,100}]

row(n) = select(x -> bigomega(x) == 2, divisors(n)); \\ Amiram Eldar, May 02 2025

A367097 Least positive integer whose multiset of prime indices has exactly n distinct semi-sums.

Original entry on oeis.org

1, 4, 12, 30, 60, 210, 330, 660, 2730, 3570, 6270, 12540, 53130, 79170, 110670, 221340, 514140, 1799490, 2284590, 4196010, 6750870, 13501740, 37532220, 97350330, 131362770, 189620970, 379241940, 735844830, 1471689660

Offset: 0

Gus Wiseman, Nov 09 2023

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
From David A. Corneth, Nov 15 2023: (Start)
Terms are cubefree.
bigomega(a(n)) = A001222(a(n)) >= A002024(n) + 1 = floor(sqrt(2n) + 1/2) + 1 for n > 0. (End)

			The prime indices of 60 are {1,1,2,3}, with four semi-sums {2,3,4,5}, and 60 is the first number whose prime indices have four semi-sums, so a(4) = 60.
The terms together with their prime indices begin:
       1: {}
       4: {1,1}
      12: {1,1,2}
      30: {1,2,3}
      60: {1,1,2,3}
     210: {1,2,3,4}
     330: {1,2,3,5}
     660: {1,1,2,3,5}
    2730: {1,2,3,4,6}
    3570: {1,2,3,4,7}
    6270: {1,2,3,5,8}
   12540: {1,1,2,3,5,8}
   53130: {1,2,3,4,5,9}
   79170: {1,2,3,4,6,10}
  110670: {1,2,3,4,7,11}
  221340: {1,1,2,3,4,7,11}
  514140: {1,1,2,3,5,8,13}

The non-binary version is A259941, firsts of A299701.
These are the positions of first appearances in A366739.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, complement A100959.
A056239 adds up prime indices, row sums of A112798.
A299702 ranks knapsack partitions, counted by A108917.
A366738 counts semi-sums of partitions, strict A366741.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A002024, A004709, A304793, A365920, A366740, A366753, A367093, A367095.

nn=1000;
w=Table[Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
v=Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

from itertools import count
from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
def A367097(n): return next(k for k in count(1) if len({sum(s) for s in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)}) == n) # Chai Wah Wu, Nov 13 2023

2 | a(n) for n > 0. - David A. Corneth, Nov 13 2023

a(17)-a(22) from Chai Wah Wu, Nov 13 2023

a(23)-a(28) from David A. Corneth, Nov 13 2023

A220423 Cubefree products of primorials (A002110).

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 210, 420, 900, 1260, 2310, 4620, 6300, 13860, 30030, 44100, 60060, 69300, 180180, 485100, 510510, 900900, 1021020, 3063060, 5336100, 6306300, 9699690, 15315300, 19399380, 58198140, 69369300, 107207100, 223092870, 290990700

Offset: 1

Reinhard Zumkeller, Dec 14 2012

nonn

Suggested by a comment of Charles R Greathouse IV in A220264.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A004709 and A073491.

Cf. A000040, A001221, A001222, A002110, A006530, A051903, A064648, A212793, A220264 (subsequence).

import Data.Set (deleteFindMin, empty, fromList, union)
import qualified Data.Set as Set (null)
a220423 n = a220423_list !! (n-1)
a220423_list = f (splitAt 1 a002110_list) empty where
   f (us'@(u:_), vs'@(v:vs)) s
     | Set.null s || m > u
                 = f (v:us', vs) (s `union` (fromList $ map (* u) us'))
     | otherwise = m : f (us', vs') s'
     where (m,s') = deleteFindMin s

A212793(a(n)) = 1; A051903(a(n)) < 3.
A001221(a(n)) <= A001222(a(n)) <= 2*A001221(a(n)).
A006530(a(n)) = A000040(A001221(a(n))).
Sum_{n>=1} 1/a(n) = (S(1)^2 + S(2))/2 = 2.093360845965235020766040..., where S(k) = Sum_{n>=0} 1/(A002110(n))^k (S(1) = 1 + A064648). - Amiram Eldar, Sep 24 2023

A367098 Number of divisors of n with exactly two distinct prime factors.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 3, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 2, 0, 1, 3, 0, 2, 1, 3, 0, 6, 0, 1, 2, 2, 1, 3, 0, 4, 0, 1, 0, 5, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 09 2023

			The a(n) divisors for n = 1, 6, 12, 24, 36, 60, 72, 120, 144, 216, 288, 360:
  .  6  6   6   6   6   6   6   6    6    6    6
        12  12  12  10  12  10  12   12   12   10
            24  18  12  18  12  18   18   18   12
                36  15  24  15  24   24   24   15
                    20  36  20  36   36   36   18
                        72  24  48   54   48   20
                            40  72   72   72   24
                                144  108  96   36
                                     216  144  40
                                          288  45
                                               72

Amiram Eldar, Table of n, a(n) for n = 1..10000

For just one distinct prime factor we have A001222 (prime-power divisors).
This sequence counts divisors belonging to A007774.
Counting all prime factors gives A086971, firsts A220264.
Column k = 2 of A146289.
- Positions of zeros are A000961 (powers of primes), complement A024619.
- Positions of ones are A006881 (squarefree semiprimes).
- Positions of twos are A054753.
- Positions of first appearances are A367099.
A001221 counts distinct prime factors.
A001358 lists semiprimes, complement A100959.
A367096 lists semiprime divisors, sum A076290.
Cf. A000005, A001248, A056170, A079275, A090885, A366740.

Table[Length[Select[Divisors[n], PrimeNu[#]==2&]],{n,100}]
a[1] = 0; a[n_] := (Total[(e = FactorInteger[n][[;; , 2]])]^2 - Total[e^2])/2; Array[a, 100] (* Amiram Eldar, Jan 08 2024 *)

a(n) = {my(e = factor(n)[, 2]); (vecsum(e)^2 - e~*e)/2;} \\ Amiram Eldar, Jan 08 2024

a(n) = (A001222(n)^2 - A090885(n))/2. - Amiram Eldar, Jan 08 2024

cp's OEIS Frontend

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

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A086971 Number of semiprime divisors of n.

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A065119 Numbers k such that the k-th cyclotomic polynomial is a trinomial.

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A366740 Positive integers whose semiprime divisors do not all have different Heinz weights (sum of prime indices, A056239).

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A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).

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A367093 Least positive integer with n more semiprime divisors than semi-sums of prime indices.

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